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Properties

Label 1-456-456.269-r0-0-0
Degree $1$
Conductor $456$
Sign $0.939 - 0.341i$
Analytic cond. $2.11765$
Root an. cond. $2.11765$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s + (0.766 − 0.642i)35-s + 37-s + (0.766 − 0.642i)41-s + (0.939 + 0.342i)43-s + (−0.173 + 0.984i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (−0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s + (0.766 − 0.642i)35-s + 37-s + (0.766 − 0.642i)41-s + (0.939 + 0.342i)43-s + (−0.173 + 0.984i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(2.11765\)
Root analytic conductor: \(2.11765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 456,\ (0:\ ),\ 0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9869048338 - 0.1736858768i\)
\(L(\frac12)\) \(\approx\) \(0.9869048338 - 0.1736858768i\)
\(L(1)\) \(\approx\) \(0.8894532276 - 0.04490964256i\)
\(L(1)\) \(\approx\) \(0.8894532276 - 0.04490964256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.64912085901855834452299537768, −23.14427333644157274098716571548, −22.67290174416075630755238493647, −21.37026662299653420281782304812, −20.416528875078116180070413767218, −19.73955285283073448330863948021, −19.01228701101963862020058318840, −17.97003720804796923557588960318, −17.13382720733105853531117386209, −16.06445677474076825247468202031, −15.381671647156711065761372559971, −14.61942863010990962283135068730, −13.28985385619123885943769840626, −12.783184404174393772585481740, −11.59074334007802746931528849598, −10.65734885830681509934179611001, −10.06971329984141249509106280837, −8.67518601539143562839749807809, −7.70355324319545260328973678596, −7.03508622275038217042206279958, −5.9362408725906715609566334252, −4.49069095327783854759611874482, −3.74920157489290622691553140855, −2.69972073670797114248704880198, −0.96679794718161018833885638944, 0.77621634538398889377311684717, 2.56456850663877354835877120565, 3.4759785441446192845452662607, 4.64385063295959605144740921142, 5.683759053700695012664101364598, 6.72069445612639427594358301177, 7.852175228953988693642812498760, 8.783680481943224370565740028602, 9.37943056675173664842835578418, 10.97678814507729318053223621309, 11.470871040220795323698530312184, 12.52540053139885149551280648642, 13.25858156266304898533637371479, 14.38404385899899743635445438388, 15.49316946515393877110744640561, 16.06461562089397903198140739850, 16.67244053077324853484777050827, 18.189444357959010011949761769013, 18.8309156245586942554684049331, 19.43301993724297076349066123627, 20.593112549554597614868178378229, 21.2349315944781731999630266418, 22.32716540526512415472483687344, 23.03754030406345633741563296019, 23.96489288720243146125399074111

Graph of the $Z$-function along the critical line